Finite Group Databasebeta

← #399⋊H(3,𝔽3) #13141249

Subgroups — #399⋊H(3,𝔽3) #13141249

Order 1728.

294 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#95⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C32×#399C311yesmaximal, characteristic, complemented
3C3×#399⋊C3C313yesmaximal, complemented
4C22×C22⋊H(3,𝔽3)—41nomaximal
5C2×C32×C4⋊C4—31no—
6C2×C32×C22⋊C4—31no—
7C32×#95C2×C311yescharacteristic, complemented
8C3×#95⋊C3C2×C313yescomplemented
9C2×C22⋊H(3,𝔽3)—42no—
10C2×C22⋊H(3,𝔽3)—41no—
11C3×#399—31no—
12C3×#399C3211yescharacteristic
13#399⋊C3—33no—
14C24×C32A411yescharacteristic
15C22×C3×A4—43no—
16C22×C32×C4—31no—
17C22×C32×C4—31no—
18C32×C4⋊C4—62no—
19C32×C22⋊C4—62no—
20C22×H(3,𝔽3)—161no—
21C22⋊H(3,𝔽3)—41no—
22C2×C3×C4⋊C4—31no—
23C2×C3×C4⋊C4—33no—
24C2×C3×C22⋊C4—33no—
25C2×C3×C22⋊C4—31no—
26C3×#95—31no—
27C3×#95C2×C3211yescharacteristic
28#95⋊C3—33no—
29C23×C32—61no—
30C23×C32—61no—
31C23×C32SL(2,𝔽3)12yes—
32C23×C32C2×A411yescharacteristic
33C2×C3×A4—46no—
34C2×C3×A4—43no—
35C2×C32×C4—32no—
36C2×C32×C4—62no—
37C2×C32×C4—61no—
38C2×C32×C4—62no—
39#399H(3,𝔽3)11yescharacteristic, complemented, Sylow
40C2×H(3,𝔽3)—162no—
41C2×H(3,𝔽3)—161no—
42C24×C3—31no—
43C24×C3C3×A411yescharacteristic
44C22×A4—123no—
45C22×C3×C4—33no—
46C22×C3×C4—31no—
47C22×C3×C4—33no—
48C22×C3×C4—31no—
49C3×C4⋊C4—62no—
50C3×C4⋊C4—66no—
51C3×C22⋊C4—62no—
52C3×C22⋊C4—66no—
53C22×C32—31no—
54C22×C32—31no—
55C22×C32—41no—
56C22×C32—62no—
57C22×C32—121no—
58C22×C32—163no—
59C22×C32C2×SL(2,𝔽3)11yescharacteristic
60C3×A4—43no—
61C32×C4—62no—
62C32×C4—121no—
63C2×C4⋊C4—31no—
64C2×C22⋊C4—31no—
65#95C2×H(3,𝔽3)11yescharacteristic, complemented
66H(3,𝔽3)—161noSylow
67C23×C3—32no—
68C23×C3—31no—
69C23×C3—61no—
70C23×C3—61no—
71C23×C3—63no—
72C23×C3—63no—
73C23×C3C3×SL(2,𝔽3)12yes—
74C23×C3C2×C3×A411yescharacteristic
75C2×A4—126no—
76C2×A4—123no—
77C2×C3×C4—36no—
78C2×C3×C4—32no—
79C2×C3×C4—62no—
80C2×C3×C4—66no—
81C2×C3×C4—63no—
82C2×C3×C4—61no—
83C2×C3×C4—66no—
84C2×C3×C4—62no—
85C2×C32—31no—
86C2×C32—31no—
87C2×C32—42no—
88C2×C32—166no—
89C2×C32—163no—
90C2×C32#127⋊C311yescharacteristic
91C24C22⋊H(3,𝔽3)11yescharacteristic
92C22×C4—31no—
93C22×C4—31no—
94C4⋊C4—62no—
95C22⋊C4—62no—
96C22×C3—31no—
97C22×C3—33no—
98C22×C3—33no—
99C22×C3—31no—
100C22×C3—31no—
101C22×C3—41no—
102C22×C3—62no—
103C22×C3—66no—
104C22×C3—121no—
105C22×C3—121no—
106C22×C3—123no—
107C22×C3—483no—
108C22×C3C2×C3×SL(2,𝔽3)11yescharacteristic
109A4—123no—
110C3×C4—66no—
111C3×C4—62no—
112C3×C4—123no—
113C3×C4—121no—
114C32—163no—
115C32#399⋊C311yescharacteristic, complemented
116C23—61no—
117C23—61no—
118C23Q8⋊H(3,𝔽3)12yes—
119C23C2×C22⋊H(3,𝔽3)11yescharacteristic
120C2×C4—32no—
121C2×C4—62no—
122C2×C4—61no—
123C2×C4—62no—
124C2×C3—33no—
125C2×C3—33no—
126C2×C3—31no—
127C2×C3—31no—
128C2×C3—31no—
129C2×C3—42no—
130C2×C3—122no—
131C2×C3—486no—
132C2×C3—483no—
133C2×C3C3×#127⋊C311yescharacteristic
134C22—31no—
135C22—31no—
136C22—41no—
137C22—62no—
138C22—121no—
139C22C2×Q8⋊H(3,𝔽3)11yescharacteristic
140C4—62no—
141C4—121no—
142C3—31no—
143C3—483no—
144C3C3×#399⋊C311yescharacteristic
145C2—31no—
146C2—31no—
147C2—42no—
148C2#127⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C3×#127⋊C3
commutator, cosocleC3×#95C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC32×#399C3
hypercenterC2×C32#127⋊C3
hypocenter#95C2×H(3,𝔽3)
LCSG ▷ C3×#95 ▷ #95
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #95⋊H(3,𝔽3)C2semidirect
2. C32×#399C3semidirect
3. C3×#399⋊C3C3semidirect
7. C32×#95C2×C3semidirect
8. C3×#95⋊C3C2×C3semidirect
39. #399H(3,𝔽3)semidirect
65. #95C2×H(3,𝔽3)semidirect
115. C32#399⋊C3semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#95⋊H(3,𝔽3)2
C32×#3993
C3×#399⋊C33
C22×C22⋊H(3,𝔽3)4